Problem: Circletown's limits form a perfectly circular shape. It has a population of $20{,}000$ and a population density of $480$ people per square kilometer. What is Circletown's radius? Round your answer, if necessary, to the nearest hundredth.
Answer: This is a density word problem. To solve it, we can use the following equation, which is the area definition of density: ${\text{Density}}=\dfrac{{\text{Total quantity}}}{{\text{Area}}}$ What do we know? There are ${20{,}000}$ people in Circletown. This is the ${\text{total quantity}}$. The population ${\text{density}}$ is ${480}$ people per square kilometer. What do we need to find? The town's radius, which gives us its ${\text{area}}$. Let's denote the town's radius as $ r$. Then, the ${\text{area}}$ is ${\pi\cdot r^2}$ square kilometers. Now we can plug ${\text{density}=480}$, ${\text{total quantity}=20{,}000}$, and ${\text{area}=\pi\cdot r^2}$ in the equation. $\begin{aligned} {\text{Density}}&=\dfrac{{\text{Total quantity}}}{{\text{Area}}} \\\\ {480}&=\dfrac{{20{,}000}}{{\pi\cdot r^2}} \\\\ {\pi\cdot r^2}\cdot{480}&=\dfrac{{20{,}000}}{\cancel{{\pi\cdot r^2}}}\cdot\cancel{{\pi\cdot r^2}} \\\\ r^2&=\dfrac{20{,}000}{480\pi} \\\\ r&=\sqrt{\dfrac{20{,}000}{480\pi}}\approx3.64 \end{aligned}$ Circletown's radius is approximately $3.64$ kilometers.